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The factorization number $F_2(G)$ of a finite group $G$ is the number of all possible factorizations of $G=HK$ as product of its subgroups $H$ and $K$, while the subgroup commutativity degree $\mathrm{sd}(G)$ of $G$ is the probability of…

Let R be a discrete valuation ring with residue field of characteristic p>0. Let K be its fraction field. We prove that any finite and flat R-group scheme, isomorphic to \mu_{p^2,K} on the generic fiber, is the kernel in a short exact…

In this paper we establish generation of analytic strongly continuous semigroup in $L^p$--spaces for the symmetric matrix Schr\"odinger operator $div(Q\nabla u)-Vu$, where, for every $x\in\mathbb{R}^d$, $V(x)=(v_{ij}(x))$ is a semi-definite…

Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac--Moody algebra $\frak{g}$ over ${\mathbb Q}$. Let $V=V^{\lambda}$ be an integrable highest weight $\frak{g}$-module and let $V_{\mathbb Z}=V^{\lambda}_{\mathbb Z}$…

The main result of this paper utilizes the representation graph of a group $G$, $R(V,G)$, and gives a general construction of a diagrammatic category $\mathbf{Dgrams}_{R(V,G)}$. The proof of the main theorem shows that, given explicit…

We consider continuous representations of the Galois group G of a number field K taking values in the completion C of an algebraic closure A of the field of l-adic numbers. We give a construction of irreducible representations of G in…

Let $N$ be a positive integer and $K$ be a number field. Suppose that $f_1,f_2 \in S_k(\Gamma_0(N))$ are two newforms such that their residual Galois representations at $2$ are isomorphic. Let $\omega_2: G_{\mathbb Q} \rightarrow {\mathbb…

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of…

Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid $M$, it is natural to ask how long a sequence of elements of $M$ needs to be to ensure the…

We define various monoid versions of the R. Thompson group $V$, and prove connections with monoids of acyclic digital circuits. We show that the monoid $M_{2,1}$ (based on partial functions) is not embeddable into Thompson's monoid ${\sf…

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…

We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we…

We are given a finite group $H$, an automorphism $\tau$ of $H$ of order $r$, a Galois extension $L/K$ of fields of characteristic zero with cyclic Galois group $\langle\sigma\rangle$ of order $r$, and an absolutely irreducible…

Let $V$ be an $n$-dimensional left vector space over a division ring $R$ and $n\ge 3$. Denote by ${\mathcal G}_{k}$ the Grassmann space of $k$-dimensional subspaces of $V$ and put ${\mathfrak G}_{k}$ for the set of all pairs $(S,U)\in…

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality…

We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by…

The Thompson-Higman groups G_{k,i} have a natural generalization to monoids M_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features of M_{k,i} and Inv_{k,i} and investigate the computational complexity of decision…

Let $V$ be a standard subspace in the complex Hilbert space $H$ and $G$ be a finite dimensional Lie group of unitary and antiunitary operators on $H$ containing the modular group $(\Delta_V^{it})_{t \in R}$ of $V$ and the corresponding…

A relationship between continued fractions and Weyl groupoids of Cartan schemes of rank two is found. This allows to decide easily if a given Cartan scheme of rank two admits a finite root system. We obtain obstructions and sharp bounds for…

Suppose that $A,B \in {\rm PSL}(2,\mathbb{R})$ generate a discrete and free group of rank 2, and let $m,n\ge 1$. We consider subgroups $\langle R,S\rangle$ of ${\rm PSL}(2,\mathbb{R})$ generated by roots of $A$ and $B$, i.e., by elements…